Demonstrations of Dirac equation covariance state:(adsbygoogle = window.adsbygoogle || []).push({});

The Dirac equation is

[itex](i γ^{μ} ∂_{μ} - m)ψ(x) = 0. \ \ \ \ \ \ \ \ \ \ [/itex] [1]

If coordinates change in a way that

[itex]x \rightarrow x' = Lx, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/itex] [2]

where [itex]L[/itex] is a Lorentz transformation, [1] should mantain its form, obtaining in the new system:

[itex](i γ^{μ} ∂'_{μ} - m)ψ'(x') = 0, \ \ \ \ \ \ \ \ [/itex] [3]

where

[itex]ψ'(x') = S(L)ψ(x) \ \ \ \ \ \ \ \ [/itex] [4]

and [itex]S(L)[/itex] is an invertible matrix rappresenting the fact [itex]ψ'(x')[/itex] should be a linear combination of [itex]ψ(x)[/itex] and should depend on [itex]L[/itex].

Remembering that from [2] stems

[itex]∂ \rightarrow ∂' = L^{-1}∂, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/itex] [5]

and substituting [4] and [5] in [3] we obtain

[itex](i γ^{μ} L^{-1 ρ}_{μ} ∂_{ρ} - m)Sψ(x) = 0. \ \ \ \ \ \ \ \ \ [/itex] [6]

Multiplying on the left for [itex]S^{-1}[/itex]:

[itex](i S^{-1} γ^{μ} L^{-1 ρ}_{μ} ∂_{ρ}S - m)ψ(x) = 0[/itex]

and because [itex]S[/itex] depends on [itex]L[/itex] that don't vary along coordinates

[itex](i S^{-1} γ^{μ} L^{-1 ρ}_{μ}S∂_{ρ} - m)ψ(x) = 0. \ \ \ \ \ \ \ \ \ [/itex] [7]

To obtain covariance [7] must be equals to [1] so

[itex]S^{-1} γ^{μ} L^{-1 ρ}_{μ}S = γ^{ρ}.\ \ \ \ \ \ \ \ \ [/itex] [8]

At this point all books state that [8] is equivalent to say

[itex]S^{-1} γ^{μ} S = L^{μ}_{ρ}γ^{ρ}.\ \ \ \ \ \ \ \ \ [/itex] [9]

This requires [itex]S[/itex] and [itex]L[/itex] to commute.

I don' understand how it comes. They are both Lorentz transformations so not necessarly commute.

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Demonstration of Dirac equation covariance

Loading...

Similar Threads for Demonstration Dirac equation |
---|

I Two particles Dirac type equation question |

I Dirac equation, one question |

I Spin Angular Momentum Dirac Equation |

I Is the Dirac free equation actually free |

I Simplify the Dirac Energy Equation? |

**Physics Forums | Science Articles, Homework Help, Discussion**